We investigate the long time behaviour for the compressible FENE dumbbell model in this section. The first difficult for us is that the stress tensor \(\tau \)  does not decay fast enough. Therefore, we failed to use the bootstrap argument as in [1]}, [2]}, [3]}. To overcome this difficulty, we need to consider the coupling effect between \(\rho \) , \(u\)  and \(g\)  in Fourier space. Motivated by [4]} and [5]}, we obtain the \(L^2\)  decay rate by taking Fourier transform in (REF ) and using the Fourier splitting method. By virtue of the standard method, one can not obtain the optimal decay rate. However, we can obtain a weaker result as follow.
